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N has 3 prime factors, N^2 has 7!! positive divisors, N^3 has 10!!! positive divisors, __ has 13!!!! positive divisors?

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N has 3 prime factors.
N^2 has 7!! positive divisors.
N^3 has 10!!! positive divisors.
___ has 13!!!! positive divisors.

posted May 24, 2017 by Pushpak Chauhan

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We are given a positive integer N. Two of its positive divisors are chosen and the differences between N and these two divisors are 270 and 280 respectively.

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See the following table

Number    Number of positive divisors
1           1
2*2         3
3*3*3       4
4*4*4*4     9 

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We call a positive integer a "good number", if the product of all its divisors equals its cube.

For example, 12 is a good number, because the divisors of 12 are 1, 2, 3, 4, 6, 12, and 1*2*3*4*6*12=1728=12^3.

If n is a good number, what is the minimum number of divisors that n^2 has?

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